Which of the following correctly identifies the transformations undergone from f(x)=|x|

Question

to f(−x)+6=|−x|+6
?(1 point)
Responses

reflected across the y-axis and translated up vertically
reflected across the y -axis and translated up vertically

reflected across the x-axis and translated down vertically
reflected across the x -axis and translated down vertically

reflected across the x-axis and translated up vertically
reflected across the x -axis and translated up vertically

reflected across the y-axis and translated down vertically

Answer 1

To analyze the transformations from \( f(x) = |x| \) to \( f(-x) + 6 = | - x| + 6 \):

1. Reflection: The function \( f(-x) = |-x| \) reflects the original function \( f(x) \) across the y-axis. Since \( |-x| = |x| \), this transformation does not change the shape of the graph but reflects it across the y-axis.

2. Vertical Translation: The addition of 6 in \( f(-x) + 6 \) translates the graph upward by 6 units.

Putting these two transformations together: the original function \( f(x) = |x| \) is first reflected across the y-axis, and then it is translated vertically upwards by 6 units.

Therefore, the correct choice is: reflected across the y-axis and translated up vertically.